By Falko Lorenz
From Math experiences: "This is a captivating textbook, introducing the reader to the classical elements of algebra. The exposition is admirably transparent and lucidly written with merely minimum necessities from linear algebra. the recent thoughts are, at the least within the first a part of the ebook, outlined within the framework of the advance of rigorously chosen difficulties. hence, for example, the transformation of the classical geometrical difficulties on structures with ruler and compass of their algebraic environment within the first bankruptcy introduces the reader spontaneously to such primary algebraic notions as box extension, the measure of an extension, etc... The booklet ends with an appendix containing routines and notes at the prior components of the ebook. notwithstanding, short historic reviews and recommendations for additional analyzing also are scattered during the text."
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Extra resources for Algebra: Volume I: Fields and Galois Theory
Clearly D O is an equivalence relation on R. F1. In an integral domain R we have a D O b if and only if there is a unit " of R such that b D "a. The simple proof is left to the reader. In the ring ޚwe have a D O b if and only if a D ˙b. But in general an integral domain has more units than just 1 and 1: Examples. (a) If R is a integral domain, so is the polynomial ring RŒX , and RŒX D R . (b) For the subring R D ޚŒi of ރwe have R D f˙1; ˙i g. (c) For a ﬁeld K we of course have K D K r f0g.
Now let be a given nonzero prime of R. The corresponding -adic valuation w W K ! [ ޚf1g in R (page 40) can be extended to a map w W KŒX ! f / is the exponent of the highest that ﬁts in all coefﬁcients of f . f / for c 2 K; f 2 KŒX : The springboard for the proof of Gauss’s Theorem is provided by the next result: F5. Let R be a unique factorization domain and ¤ 0 a prime in R. h/: Proof. Clearly, for every f 2 KŒX there exists c 2 R such that cf 2 RŒX . So taking (7) into account, we can assume without loss of generality that g; h 2 RŒX .
Proof. (a) The intersection of all subﬁelds of K is a subﬁeld of K. It is the smallest subﬁeld of K, hence a prime ﬁeld. (b) Let K be any ﬁeld and K0 its prime ﬁeld. Clearly, ޚK Â K0 . Now, in case A above, ޚK is itself already a subﬁeld of K, so K0 D ޚK ' = ޚp ޚ. In case B we have K0 D Frac ޚK ' Frac ޚD ޑ. ˜ Remarks. (a) It is customary to write just n instead of nK , and we will do so. But you should keep an eye open in each case for whether the n represents an integer or an element of K.